Evaluation of land resources carrying capacity based on entropy weight and cloud similarity

Land is the foundation of human life and development, which is also the most important part of a country. The study of land carrying capacity is one of the important contents of land management, wherein the evaluation of land resource carrying capacity (LRCC) is an important reference for land resource planning. Aiming at the information fuzziness and uncertainty in the evaluation of LRCC, firstly, a comprehensive evaluation model based on entropy weight and normal cloud similarity was proposed, which is based on cloud model theory and combined with normal cloud similarity measurement method and entropy weight method. Secondly, taking the asphalt pavement experiment as an example for empirical analysis, the experimental results are consistent with the actual situation, which proves the feasibility and effectiveness of the proposed model. Finally, taking China’s Chongqing city as the research area, the proposed evaluation model is used to study LRCC. The research results indicate that the comprehensive carrying capacity and average carrying capacity of various systems in China’s Chongqing have been improved in the past decade. Among them, the comprehensive carrying capacity rose from the second level during the "12th Five-Year Plan" period to the third level during the "13th Five-Year Plan" period. In the future, it is necessary to focus on the improvement of soil and water resources system and economic and technological system. This conclusion reflects LRCC of Chongqing in China objectively and has a reference value for Chongqing's land planning.


Cloud model concept and characteristic curve
Definition 1. 20 Let U be a universal set described by precise numbers, and C be the qualitative concept related to U. If there is a number x ∈ U , which randomly realizes the concept C, and the membership degree of x for C, that is, µ C (x) ∈ [0, 1][0, 1] , is a random value with steady tendency: then the distribution of x on U is s defined as a cloud, and each x is defined as a cloud drop, noted Drop(x, µ C (x)).
The cloud model generally consists of three numerical characteristics (expectation Ex , entropy En and hyper- entropy He ) to describe the uncertainty information as a whole, where Ex reflects the central value of uncertainty information; En reflects the degree of discreteness of the data to the expected value Ex , and represents the range of the data, that is, reflects the ambiguity of the data; hyper-entropy ( He ) is the entropy of entropy(En ), which represents the range of random distribution of cloud droplets, reflects the randomness of data, and indicates the degree of discreteness of cloud droplets.
If the distribution of x on U satisfies: x ∼ N(Ex, En ′2 ) , where En ′ ∼ N(En, He 2 ) , and the degree of certainty on C is: Then the distribution of x on U is called normal cloud.The expectation curve of normal cloud with entropy was defined by literature 17 based on the conclusion that the expectation of normal cloud droplet is Ex , variance is En 2 + He 2 given in literature 21 .
Definition 2 17 If the random variable x ∼ N(Ex, En ′2 ) , where En ′ ∼ N(En, He 2 ) , and En = 0 , then is called the entropy-containing expectation curve of a normal cloud.

Introduction to Wasserstein distance
Definition 3 22 Let µ,v be the measures on the probability space ℜ n , ∀x, y ∈ ℜ n , define the p-Wasserstein distance as where (µ, v) is the set of joint probability measures γ on ℜ n × ℜ n , the edge distribution of this joint probability distribution are µ and v , d(x, y) is any distance on ℜ l , p ≥ 1. Definition 4 23 For two multidimensional normal distributions P 1 and P 2 , the Wasserstein distance is: where m 1 and m 2 are the mean vectors of P 1 and P 2 respectively, M 1 and M 2 are the covariance matrices of P 1 and P 2 respectively.
According to formula (4), if for two one-dimensional normal distributions X and Y , the Wasserstein distance d(X, Y ) between the two is.
where µ 1 and µ 2 are the mean of X and Y respectively, σ 2 1 and σ 2 2 are the variances of X and Y respectively.

Normal cloud similarity calculation method based on Wasserstein distance
Through the above research, Wasserstein distance and normal cloud similarity are combined to obtain the method of normal cloud similarity based on Wasserstein distance. (2) The larger the sim(C 1 , C 2 ) , the higher the similarity of the two normal clouds, and the opposite is true.Meanwhile, sim(C 1 , C 2 ) also satisfies the fol- lowing properties.

Evaluation model
The determination of weights is one of the important factors in comprehensive evaluation.Entropy weight can objectively reflect the weight of indicators and eliminate human interference in the weight of each indicator, thereby making the results more realistic.Based on the advantages of normal cloud similarity in comprehensive evaluation, a comprehensive evaluation model based on entropy weight and normal cloud similarity is proposed by utilizing the characteristics of cloud models in dealing with fuzziness and uncertainty, and combining the objectivity and adaptability advantages of entropy weight method.

Entropy weight method
Entropy weight method is an objective weighting method.The weight is determined by the information of each index, which can avoid the deviation caused by human subjective factors.The main steps are as follows 25 : 1.According to the initial data set of n evaluation indexes of m evaluation objects, and established the eigenvalue matrix X: X = (x ij ) m×n (i = 1, 2, ..., m; j = 1, 2, ..., n). 2. As the dimensions and orders of magnitude of each index are different, it is necessary to standardize the matrix.The processing formulas of positive and negative indicators are as follows: In the above formula, x min j and x max j are respectively the minimum and maximum values of the same index x j in different objects.The normalized matrix is R = (r ij ) m×n (i = 1, 2, ..., m; j = 1, 2, ..., n).

The entropy of each evaluation index is defined as:
where 2. The entropy weight w j of the j evaluation index is: where

Construction of evaluation model
According to 2.3 and 3.1, this paper proposes a comprehensive evaluation model based on entropy weight and normal cloud similarity.The steps as follows: Step 1: Establish the factor domain U = {u 1 , u 2 ...., u n } of the evaluation object and the comment domain Step 2: Use entropy weight method to calculate the index weight W = {w 1 , w 2 ...., w i }.
Step 3: Build a cloud model of the evaluation object.According to the original data under each index i , use the backward cloud generator to obtain the cloud concept A i = (Ex i , En i , He i ) i = 1, 2, ..., n under each indicator i.
Step 4: Construct the classification level of the evaluation object.Let the upper and lower boundary values of level j(j = 1, 2, ..., m) corresponding to index i(i = 1, 2, ..., n) be x 1 ij and x 2 ij , then the qualitative concept of level j corresponding to index i can be represented by normal cloud concept, where 26 : As x 1 ij and x 2 ij are critical values from one evaluation level interval to another evaluation level interval, they are boundary values with randomness and fuzziness, so x 1 ij , x 2 ij can belong to two adjacent evaluation levels at the same time.Thus, x 1 ij , x 2 ij has the same membership degree in the two adjacent evaluation levels, namely: Hyper-entropy He ij is the entropy of entropy, which determines the degree of dispersion between cloud droplets.The larger the value is, the better the cohesiveness between cloud droplets will be.When the value decreases to 0, the normal cloud concept will degenerate into a normal cloud curve.Hyper-entropy is generally obtained through experience.
Step 5: According to the similarity calculation formula (formula ( 6)), calculated the similarity between the cloud concept A i = (Ex i , En i , He i ) of each index i and the evaluation level, and obtained the normal cloud similarity matrix Z = (z ij ) n×m .
Step 6: Weight set W = {w 1 , w 2 ...., w i } and normal cloud similarity matrix Z = (z ij ) n×m are used for aggrega- tion to obtain the comprehensive similarity matrix D at each level. where According to the principle of maximum similarity, the j evaluation level corresponding to the maximum similarity is selected as the result of comprehensive evaluation.
The evaluation process is shown in Fig. 1.

Validation analysis
In order to verify the feasibility and effectiveness of the proposed method, this section will analyze the similarity measurement method and the comprehensive evaluation model based on entropy weight and normal cloud similarity.In terms of similarity, the classification accuracy and time complexity of WCM have been verified and explained in literature 24 , which will not be elaborated here.In terms of the model, this paper selects the asphalt pavement performance evaluation experiment 27 as an example for simulation experiments, and compares the results with the original text to demonstrate the accuracy of the model.Select Xinjiang's S215 Line Sanchakou-Shache Expressway 233 km in length, and with the K81-K86 total 5 km as the test section.Taking the test data in 2016 as the sample and shown in Table 1, establish the entropy weight and normal cloud similarity concept, select five evaluation indexes, and the index evaluation levels were shown in Table 2 27 .
Because the evaluation indicators are positive indicators, Calculate the weight of each evaluation index from Table 1 and formula (7), (9).Meanwhile, according to Table 1, cloud concept A i = (Ex i , En i , He i ) corresponding to each index is generated through backward cloud generator.The results are shown in Table 3.

Entropy weight method
Backward cloud generator ( , , ) Ex En He The numerical characteristic values (Ex, En, He) of each evaluation index are obtained by formula (10) and formula (11), The results are shown in Table 4.
According to formula (6), calculate the normal cloud similarity between cloud concept A i corresponding to each index in Table 3 and each evaluation level in Table 4.The normal cloud similarity matrix is shown in Table 5.     www.nature.com/scientificreports/Finally, the weight of each index and the normal cloud similarity matrix are used for aggregation, and the comprehensive similarity matrix D is calculated according to formula (12).The results are shown in Table 6.
It can be seen from Table 6 that the maximum value of the comprehensive similarity corresponds to the level of high.According to the principle of maximum similarity, the evaluation level of this section is high.This conclusion is consistent with the results in literature 27 and the actual situation.In the original literature, the entropy weight method is also used to determine the index weight to ensure relative objectivity.The difference between this paper and literature 27 is that comprehensive similarity is used in this paper to determine the evaluation level, while comprehensive certainty is used in the original literature, but the results obtained are consistent.This fully demonstrates the accuracy and effectiveness of the comprehensive evaluation model proposed in this paper based on entropy weight and normal cloud similarity.

Case analysis
After the successful establishment of the comprehensive evaluation model, this paper applies the model to the evaluation of Chongqing's LRCC.The application characteristics of the model are illustrated by practical cases.Meanwhile, the evaluation of LRCC in Chongqing can also provide reference for the local land planning and development.
As the foundation of human development, land provides a variety of resources for human development and guarantees the progress of human society.In recent decades, the explosive growth of population has brought about the overdevelopment of land and the sharp consumption of resources, and land resources are facing unprecedented pressure.With the increasingly severe situation of land resources, the evaluation of LRCC has been developed rapidly.In the 12th and 13th Five-Year plans for national development, ecological and environmental protection has been given an important position, and it is proposed to continuously strengthen ecological and environmental protection and improve ecological and environmental quality.In the "13th Five-Year Plan", the construction of ecological civilization is elevated to the national strategy, highlighting the status of ecological civilization construction.LRCC refers to the limit of the scale and intensity of various human activities that land resources can carry in a certain period, a certain spatial area and under certain economic, social, resource and environmental conditions [28][29][30] .
Chongqing is located in the southwest of China, upstream of the Yangtze River.The permanent population is over 30 million, and the terrain is mainly hilly and mountainous.Chongqing is a municipality directly under the central government, a national central city, and a core city of the Chengdu Chongqing Economic Circle in China.It holds an important strategic position in the southwest region and throughout the country.This section takes Chongqing as the research area and uses the "12th Five Year Plan" and "13th Five Year Plan" as time standards to evaluate the comprehensive LRCC and the carrying capacity of various systems in Chongqing at the standard layer.This study chooses China's Chongqing as the research area, examining its geographical and strategic location, attempting to reveal the patterns of population, resources, and environment changes in Chongqing, and providing relevant references for its subsequent development.The 12th Five-Year Plan (2011-2015) is taken as an example for analysis.According to the data in Table 9, the cloud concept corresponding to each index is generated through backward cloud generator.The results are shown in Table 11.
According to formula (6), the similarity between the normal clouds corresponding to each index and the normal clouds of each evaluation level is calculated, and the similarity matrix shown in Table 12.
The weight of each index during 2011-2015 and the normal cloud similarity matrix are used for aggregation.According to formula (12), the comprehensive similarity matrix is calculated.The results are shown in Table 13.
According to the principle of maximum similarity, it is concluded that the comprehensive LRCC of Chongqing during the 12th Five-Year Plan period (2011-2015) is II, that is, the low carrying capacity level.Similarly, the comprehensive similarity matrix of the 13th Five-Year Plan period (2016-2020) can be obtained by using the same method.The results are shown in Table 14.
Also, according to the principle of maximum similarity, it is concluded that LRCC in Chongqing during 2016-2020 is level III, that is the medium carrying capacity level.The conclusion indicates that Chongqing's LRCC has been improved during the 13th Five-Year Plan period, and Chongqing has considered the protection of ecological environment while developing economy.
The data in Tables 15 and 16 were visualized to compare LRCC levels of each system in the two-time stages, where the Water and soil resource system was abbreviated as Water, Ecological environment system was abbreviated as Ecological, Social and cultural system was abbreviated as Social, Economic and technological system was abbreviated as Economic.As shown in Fig. 2.
From Fig. 2, during the "12th Five-Year Plan" and "13th Five-Year Plan" period, in terms of water and soil resources system, the carrying capacity level was level III, that is the medium carrying capacity level.This indicates that the system carrying capacity of water and soil resources in Chongqing has been maintained at a relatively stable level in the past ten years.With the growth of Chongqing's population, the intensity of land development is also increasing, which will have a certain impact on the comprehensive LRCC.In terms of the ecological environment system, the carrying capacity during the "12th Five-Year Plan" period is level IV, that is, the high carrying capacity level, while that of the "13th Five-Year Plan" period is level V, that is, the very high carrying capacity level.This change indicates that the ecological environment of Chongqing has development in the past five years.From the perspective of indicators, the forest coverage rate and air quality excellence rate are constantly improving, which have a positive impact on carrying capacity.As far as the environment is concerned, Chongqing is located in southwest China and the upper reaches of the Yangtze River.Most of its administrative areas are hilly and mountainous.Therefore, it has natural advantages in urban greening and well protected ecological environment system.In terms of social and humanistic system, the carrying capacity during the "12th  Five-Year Plan" and "13th Five-Year Plan" are both level IV, that is, the high carrying capacity level.In recent years, with the rapid development of economy, the urbanization rate of Chongqing is constantly increasing, and the unemployment rate and the natural growth rate of population are declining.These factors are affecting the change of social carrying capacity.In terms of economic and technological system, it has been uplevel from the low level in the 12th Five-Year Plan to the medium level in the 13th Five-Year Plan, which indicates that Chongqing's economy has been effectively developed in these five years.Since the Western Development and the construction of the Chengdu Chongqing Economic Circle, Chongqing's economy has developed rapidly, and people's income levels have also improved.The improvement of these indicators directly drives the improvement of the carrying capacity of the economic and technological system.In general, while developing economy, Chongqing still takes the protection of ecological environment.To further improve LRCC in the future, Chongqing can start from soil and water resources system, economic and technological system, and make efforts to prepare land planning and economic construction.

Conclusions
In this paper, a comprehensive evaluation model based on entropy weight and normal cloud similarity is proposed, which based on the cloud model and the objective characteristics of entropy weight method.The empirical analysis is carried out by taking the asphalt pavement experiment as an example.The conclusion is consistent with the original literature and the actual situation, that shows the feasibility and effectiveness of the proposed method.Finally, the model is applied to the evaluation of Chongqing's LRCC from 2011 to 2020, and the comprehensive carrying capacity and the system carrying capacity of Chongqing are analyzed.The research shows that the comprehensive LRCC of Chongqing has been improved from level II to level III.The bearing capacity of each system has also been improved.Relatively speaking, the land and water resources system, and the economic and technological system still need to be further developed.This study is realistic and objective, and can provide some reference for Chongqing's future land use planning.

Discussion
This paper still has some shortcomings and needs improvement in the following aspects, which need to be further explored in the subsequent research: 1.The construction of the index system can add more evaluation indicators and make the evaluation system more diversified.2. When determining index weights, subjective methods such as analytic hierarchy process and expert scoring can be combined with objective methods to combine and assign weights, so as to make the determination of weights more comprehensive and scientific, and further comprehensively evaluate the carrying capacity of land resources.

Figure 1 .
Figure 1.Flow chart of comprehensive evaluation based on entropy weight and normal cloud similarity.

Figure 2 .
Figure 2. The carrying capacity levels of each system during the 12th Five-Year Plan and 13th Five-Year Plan periods.

Table 2 .
Performance evaluation index of asphalt pavement of expressway.

Table 3 .
Weight of indicators and corresponding cloud concept of this section.

Table 4 .
Digital characteristics of cloud concept for evaluation levels of each index of asphalt pavement.

Table 9 .
Evaluation index data of Chongqing from 2011 to 2020.

Table 10 .
Weight of indicators during the 12th Five-Year Plan and 13th Five-Year Plan of Chongqing.

Table 11 .
Cloud concepts corresponding to each index from 2011 to 2015.

Table 15 .
Comprehensive similarity matrix of criterion layer from 2011 to 2015.

Table 16 .
Comprehensive similarity matrix of criterion layer from 2016 to 2020.